If a function $f(x)$ is defined in $[-L,L]$, then Fourier series is defined as $$f(x)=\frac{A_0}{2}+\sum_{n=1}^{\infty}[A_n \cos (n\pi x/L)+ B_n \sin (n \pi x/L)],$$ where $$A_n=L^{-1}\int_{-L}^{L} f(x) \cos (n\pi x/L)~ dx,\quad B_n=L^{-1}\int_{-L}^{L} f(x) \sin (n\pi x/L)~ dx.$$
I would like to know why we have half of $A_0$, here,